The signal intensity generated during magnetic resonance imaging can be encoded using various mechanisms to allow conversion to spatial images. For example, gradient coils (high power electromagnets) may be used to encode spatial information. The spatial encoding is achieved by causing the gradient coils to produce a linearly varying magnetic field with position in an imaging volume within which the image to be scanned is placed. In real world implementations, the field profiles produced by the gradient coils deviate from strictly linear. The amount of deviation depends on spatial position within the image.
These gradient coil non-linearities can cause errors in various types of MR imaging, including diffusion weighted, phase-contrast, or intravoxel incoherent motion (IVIM) imaging. The non-uniformity of the gradient fields often leads to spatially dependent errors in the direction and magnitude of the motion sensitive encoding. Such gradient coil non-linearities may be particularly pronounced when the gradient coils are asymmetric, for example, when imaging the brain using diffusion weighted imaging.
In order to correct for these errors, the non-linearity tensor for the gradient coil set are often calculated on a pixel by pixel basis. To calculate the non-linearity tensor, the gradient of the magnetic field produced by each gradient coil must be known. The magnetic field of the gradient coils have conventionally been found experimentally or by approximation using a spherical harmonic expansion. The gradient of these fields were then calculated numerically or analytically.
However, finding the magnetic field of the gradient coils experimentally may be problematic because 1) voxel discretization of an image leads to a discrete distortion map rather than a desirable continuous one, and 2) the distortion caused by B0 inhomogeneity from the primary magnet, which is sequence specific, is included in the distortion map.
Numerically calculating the gradient of the field may also be not ideal since one must have a sampling of the field at a higher resolution to determine what is required for the gradient. One must also sample at least two points and do a difference per direction in the calculations. This tends to lead to increased processing time.
In particular, spherical harmonic representation of the fields and analytic calculation of the gradient of the fields can pose problems for asymmetric gradient coils, which may be used for stroke imaging. The number of harmonic terms needed to accurately represent the field grows substantially for asymmetric gradient coils. Therefore, it is often difficult to express or represent such coils using spherical harmonics.